A117301 - OEIS

%I #25 Sep 12 2022 04:24:50

%S -1,-2,-12,-24,-12,-24,56,-78,-48,42,-184,-24,152,46,-260,-48,102,

%T -304,110,126,-60,276,-250,-630,-24,-12,-24,1272,-72,-1156,-294,476,

%U -24,-676,580,-374,-60,286,-740,644,-24,-1206,-12,1520,1942,-1880

%N a(n) = prime(n+3)*prime(n) - prime(n+1)*prime(n+2).

%C The number of negative values in this sequence appears to be consistently larger than the number of positive values. The following list gives the number of positive terms among the first n terms divided by the number of negative terms among the first n terms for various n.

%C      n  ratio

%C   10^2  0.51515151515...

%C   10^3  0.70940170940...

%C   10^4  0.80212650928...

%C   10^5  0.83826908582...

%C   10^6  0.86339454584...

%C Cino Hilliard conjectures that this ratio converges and that there are infinitely many elements in the sequence whose absolute value is 12.

%C It appears that the positions of negative multiples of 12 are given by A064026(n+1) for n >= 1.  If so, then Hilliard's conjecture is true, and a further conjecture is that if k >= 2 then there are infinitely many multiples of -12*k in this sequence. - _Clark Kimberling_, Jan 01 2014

%F a(n) = A090090(n) - A006094(n+1). - _Michel Marcus_, Oct 07 2013

%e a(4) = prime(4)*prime(7) - prime(5)*prime(6) = 7*17 - 11*13 = -24.

%t Table[Prime[n]*Prime[n + 3] - Prime[n + 1]Prime[n + 2], {n, 1, 100}] (* _Stefan Steinerberger_, Jun 27 2007 *)

%t (* The following program is significantly faster: *)

%t (First[#]Last[#]-#[[2]]#[[3]])&/@Partition[Prime[Range[50]],4,1] (* _Harvey P. Dale_, May 08 2011 *)

%o (PARI) det2cont(n) = {local(m,p,x, D); m=0; p=0; for(x=1,n, D=prime(x)*prime(x+3)-prime(x+1)*prime(x+2); if(D<0,m++,p++); print1(D",") ); print(); print("neg= "m); print("pos= "p); print("pos/neg = "p/m+.) }

%K sign

%O 1,2

%A _Cino Hilliard_, Apr 24 2006

%E Edited by _Stefan Steinerberger_, Jun 27 2007